Portfolio management and asset allocation require the acquisition or liquidation of positions. When the related volume is sizeable according to prevailing market conditions, placing an order is potentially able to change the price of that asset. This is particularly true for actions taken by institutional investors (e.g. pension funds or insurance companies managing large capitals) and for illiquid assets. The interaction between market participants may determine the creation of positions with the hope to profit from being on the other side of the large order. By the same token, large orders may need so–called price concessions in order to attract an adequate counterparty. The decision to buy or sell an asset in large quantities, in other words, must be informed as of the potential price impact which that particular trade may have (an effect known as slippage). This may result in lower profits or higher losses if the order is executed (transaction risk) or in the order not being executed at all.

In recent years, and increasingly so, services are being offered by specialized firms which provide program trading to institutional investors under the premise that their expertise will translate into a more efficient management of the transactions, minimizing slippage, or even into the assumption of some transaction risk. To be clear, there is no easy solution to transaction risk: the uncertainty around the actual execution price relative to one’s own expected price (or of the execution of the order itself) must be weighed against the unavoidable uncertain market movements to face, should one decide to wait to place the order (market risk). To this extent, the relevant strategy is to plan how to place the orders relative to the characteristics of the financial market (rules and regulations, e.g. opening price formation), of the particular asset (e.g. liquidity, volatility, etc.), and, at a more advanced level, of that asset relative to other assets in a portfolio (e.g. correlation, common features, etc.).

Algorithmic trading (a.k.a. algo–trading) is widely used by investors who want to manage the market impact of exchanging large amounts of assets. It is favored by the development and diffusion of computer–based pattern recognition, so that information is processed instantaneously and action is taken accordingly with limited (if any) human judgment and intervention. The size of orders generated and executed by algo–trading is quite large and is increasing. In October 2006, the NYSE has boosted a mixed system of electronic and face–to–face auction which brings automated trades to about 50% of total trades, and similar trends are valid for other financial markets (smaller proportions when assets are more complex, e.g. options). It is generally recognized that algorithmic trading has reduced the average trade size (smaller liquidity) in the markets and hence has pushed institutional investors to split their orders in order to seek better price execution (cf. Chordia et al. (2008)).

The daily Volume Weighted Average Price (VWAP) was introduced by Berkowitz et al. (1988)) as a weighted average (calculated at the end of the day) of intra-daily transaction prices with weights equal to the relative size of the corresponding traded volume to the total volume traded during the day (defined as full VWAP in Madhavan (2002)). In the original paper, the difference between the price of a trade and the recorded VWAP was used to measure the market impact of that trade. The goal of institutional investors is to minimize such impact. VWAP is a very transparent measure, easily calculated at the end of the day with tick–by–tick data: it allows to evaluate how favorable average traded prices were to the trader. A VWAP replicating strategy is thus defined as a procedure for splitting a certain number of shares into smaller size orders during the day, which will be executed at different prices with the net result of an average price that is close to the VWAP. An interesting feature of this type of strategies is that accurate intra-daily volume proportions forecasting leads to accurate VWAP replication. Whether the VWAP benchmark is proposed on an agency base or on a guaranteed base (in exchange for a fee) is a technical aspect which does not have any bearings in what we discuss.

This paper deals with volume forecasting for VWAP trading. The trade to be executed is treated as exogenously determined (cf. Bertsimas and Lo (1998), Almgren and Chriss (2000), Engle and Ferstenberg (2007)). In order to implement the replicating strategy, we assume that we are price takers and no effort will be put in predicting prices while we concentrate on modeling volumes and predicting intra-daily volume proportions. As we will show in what follows, there are different components in the dynamics of traded volumes recorded at intra-daily intervals (relative to outstanding shares). We concentrate on single assets and we record intra-daily behavior at regular intervals. From an initial descriptive analysis of the series we derive some indications as of what features the model should reproduce. Beside the well documented U-shaped pattern of intra-daily trading activity which translates into a periodic component, we find that there are two other components which relate to a daily evolution of the volumes and to intra-daily non–periodic dynamics, respectively. We use these findings as a guideline to specify an extension of the Multiplicative Error Model (Engle (2002)) called a Component Multiplicative Error Model (CMEM) where each element has its own dynamic specification. The model is specified in a semiparametric fashion, thus avoiding the choice of a specific distribution of the error term. We estimate all the parameters at once by Generalized Method of Moments. The estimated model can then be used to dynamically forecast intra-daily volumes proportions. To our knowledge there is no well established methodology to evaluate proportion forecasts. In this work we introduce a loss functions, the Slicing Loss function, for the evaluation of proportions forecasts which retains both an operational and information theoretic interpretation.

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